An efficient algorithm for finding the CSG representation of a simple polygon
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Modeling two-dimensional and three-dimensional objects is an important theme in computer graphics. Two main types of models are used in both cases: boundary representations, which represent the surface of an object explicitly but represent its interior only implicitly, and constructive solid geometry representations, which model a complex object, surface and interior together, as a boolean combination of simpler objects. Because neither representation is good for all applications, conversion between the two is often necessary.
We consider the problem of converting boundary representations of polyhedral objects into constructive solid geometry (CSG) representations. The CSG representations for a polyhedronP are based on the half-spaces supporting the faces ofP. For certain kinds of polyhedra this problem is equivalent to the corresponding problem for simple polygons in the plane. We give a new proof that the interior of each simple polygon can be represented by a monotone boolean formula based on the half-planes supporting the sides of the polygon and using each such half-plane only once. Our main contribution is an efficient and practicalO(n logn) algorithm for doing this boundary-to-CSG conversion for a simple polygon ofn sides. We also prove that such nice formulae do not always exist for general polyhedra in three dimensions.
- J. Boyse and J. Gilchrist. GMSolid: Interactive modeling for design and analysis of solids.IEEE Comput. Graphics and Applications, 2:86–97, 1982.
- C. Brown. PADL-2: A technical summary.IEEE Comput. Graphics and Applications, 2:69–84, 1982.
- M. H. Brown.Algorithm Animation. ACM Distinguished Dissertations. MIT Press, Cambridge, MA, 1988.
- M. H. Brown. Zeus: A system for algorithm animation. InProceedings of the 1991 IEEE Workshop on Visual Languages, pages 4–9, October 1991.
- M. H. Brown and J. Hershberger. Color and sound in algorithm animation.IEEE Trans, Comput., 25(12):52–63, 1992.
- B. M. Chazelle. Computational Geometry and Convexity. Technical Report CMU-CS-80-150, Department of Computer Science, Carnegie-Mellon University, Pittsburgh, PA, 1980.
- H. Edelsbrunner.Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, volume 10. Springer-Verlag, Berlin, 1987.
- W. Franklin. Polygon properties calculated from the vertex neighborhoods. InProceedings of the 3rd ACM Symposium on Computational Geometry, pages 110–118, 1987.
- R. L. Graham and F. F. Yao. Finding the convex hull of a simple polygon.J. Algorithms,4:324–331, 1983.
- L. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. Tarjan. Linear time algorithms for visibility and shortest path problems inside triangulated simple polygons.Algorithmica, 2:209–233, 1987. CrossRef
- L. Guibas, L. Ramshaw, and J. Stolfi. A kinetic framework for computational geometry. InProceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 100–111, 1983.
- J. Hershberger and M. H. Brown. Boolean Formulae for Simple Polygons. Research Report Videotape 50b, DEC Systems Research Center, Palo Alto, CA, 1989.
- K. Hoffmann, K. Mehlhorn, P. Rosenstiehl, and R. E. Tarjan. Sorting Jordan sequences in linear time. InProceedings of the ACM Symposium on Computational Geometry, pages 196–203, 1985.
- D. T. Lee. On finding the convex hull of a simple polygon.Internat. J. Comput. Inform. Sci., 12:87–98, 1983.
- M. M. Mano.Digitial Logic and Computer Design. Prentice-Hall, Englewood Cliffs, NJ, 1979.
- M. Mäntylä.An Introduction to Solid Modeling. Computer Science Press, Rockville, MD, 1987.
- D. McCallum and D. Avis. A linear algorithm for finding the convex hull of a simple polygon.Inform. Process. Lett., 9:201–206, 1979.
- A. Melkman. On-line construction of the convex hull of a simple polyline.Inform. Process. Lett., 25:11–12, 1987.
- M. Mortenson.Geometric Modeling. Wiley, New York, 1985.
- R. Newell. Solid modelling and parametric design in the Medusa system. InComputer Graphics '82, Proceedings of the Online Conference, pages 223–235, 1982.
- J. O'Rourke.Art Gallery Theorems and Algorithms. Oxford University Press, Oxford, 1987.
- M. Paterson and F. Yao. Binary partitions with applications to hidden-surface removal and solid modelling. InProceedings of the 5th ACM Symposium on Computational Geometry, pages 23–32, 1989.
- T. Pavlidis. Analysis of set patterns.Pattern Recognition, 1:165–178, 1968.
- D. Peterson. Halfspace Representation of Extrusions, Solids of Revolution, and Pyramids. SANDIA Report SAND84-0572, Sandia National Laboratories, Albuquerque, NM, 1984.
- F. P. Preparata and M. I. Shamos.Computational Geometry. Springer-Verlag, New York, 1985.
- A. Requicha. Representations for rigid solids: Theory, methods, and systems.ACM Comput. Surveys, 12:437–464, 1980.
- A. A. Schäffer and C. J. Van Wyk. Convex hulls of piecewise-smooth Jordan curves.J. Algorithms, 8:66–94, 1987.
- W. Tiller. Rational B-splines for curve and surface representation.IEEE Comput. Graphics and Applications, 3, 1983.
- S. B. Tor and A. E. Middleditch. Convex decomposition of simple polygons.ACM Trans. Graphics, 3(4):244–265, 1984.
- H. Voelcker, A. Requicha, E. Hartquist, W. Fisher, J. Metzger, R. Tilove, N. Birrell, W. Hunt, G. Armstrong, T. Check, R. Moote, and J. McSweeney. The PADL-1.0/2 system for defining and displaying solid objects.ACM Comput. Graphics, 12(3):257–263, 1978.
- J. R. Woodwark and A. F. Wallis, Graphical input to a Boolean solid modeller. InProceedings of CAD 82, pages 681–688, Brighton, U.K., 1982.
- An efficient algorithm for finding the CSG representation of a simple polygon
Volume 10, Issue 1 , pp 1-23
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Constructive solid geometry
- Computational geometry
- Boundary representation
- Monotone boolean formulae
- Incremental convex hull
- Industry Sectors
- Author Affiliations
- 1. Department of Computer Science, Princeton University, 08544, Princeton, NJ, USA
- 2. Stanford University, 94305-4055, Stanford, CA, USA
- 3. DEC Systems Research Center, 130 Lytton Avenue, 94301, Palo Alto, CA, USA
- 4. Department of Computer Science, University of British Columbia, Vancouver, BC, Canada